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Power and Effect Size Cal State Northridge 320 Andrew Ainsworth PhD

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2 Major Points Review What is power? What controls power? Effect size Power for one sample t Power for related-samples t Power for two sample t Psy 320 - Cal State Northridge

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3 Important Concepts Concepts critical to hypothesis testing –Decision –Type I error –Type II error –Critical values –One- and two-tailed tests Psy 320 - Cal State Northridge

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4 Decisions When we test a hypothesis we draw a conclusion; either correct or incorrect. –Type I error Reject the null hypothesis when it is actually correct. –Type II error Retain the null hypothesis when it is actually false. Psy 320 - Cal State Northridge

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5 Type I Errors Null Hypothesis really is true We conclude the null is false. This is a Type I error –Probability set at alpha ( ) usually at.05 –Therefore, probability of Type I error =.05 Psy 320 - Cal State Northridge

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6 Type II Errors The Alternative Hypothesis is true We conclude that the null is true This is also an error (Type II) –Probability denoted beta ( ) We can’t set beta easily. We’ll talk about this issue later. Power = (1 - ) = probability of correctly rejecting false null hypothesis. Psy 320 - Cal State Northridge

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7 Confusion Matrix Psy 320 - Cal State Northridge

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8 Critical Values These represent the point at which we decide to reject null hypothesis. e.g. We might decide to reject null when (p|null) <.05. –In the null distribution there is some value with p =.05 –We reject when we exceed that value. –That value is called the critical value. Psy 320 - Cal State Northridge

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9 One- and Two-Tailed Tests Two-tailed test rejects null when obtained value too extreme in either direction –Decide on this before collecting data. One-tailed test rejects null if obtained value is too low (or too high) –We only set aside one direction for rejection. Psy 320 - Cal State Northridge

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10 One- & Two-Tailed Example One-tailed test –Reject null if IQPLUS group shows an increase in IQ Probably wouldn’t expect a reduction and therefore no point guarding against it. Two-tailed test –Reject null if IQPLUS group has a mean that is substantially higher or lower. Psy 320 - Cal State Northridge

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11 What Is Power? Probability of rejecting a false H 0 Probability that you’ll find difference that’s really there 1 - , where = probability of Type II error Psy 320 - Cal State Northridge

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12 What Controls Power? The significance level ( ) True difference between null and alternative hypotheses 1 - 2 Sample size Population variance The particular test being used Psy 320 - Cal State Northridge

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13 Distributions Under 1 and 0 Psy 320 - Cal State Northridge

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14 Effect Size The degree to which the null is false –Depends on distance between and –Also depends on standard error (of mean or of difference between means) Psy 320 - Cal State Northridge

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15 What happened to n? It doesn’t relate to how different the two population means are. It controls power, but not effect size. We will add it in later. Psy 320 - Cal State Northridge

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16 Estimating Effect Size Judge your effect size by: –Past research –What you consider important –Cohen’s conventions Psy 320 - Cal State Northridge

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17 Combining Effect Size and n We put them together and then evaluate power from the result. General formula for Delta –where f (n) is some function of n that will depend on the type of design Psy 320 - Cal State Northridge

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18 Power for One-Sample or Related samples t First calculate delta with: –where n = size of sample, and and as above Look power up in table using and significance level ( ) Psy 320 - Cal State Northridge

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19 Power for Single Sample IQPLUS Study One sample z and t –Compared IQPLUS group with population mean = 100, sigma = 10 –Used 25 subjects –We got a sample mean of 106 and s = 7.78 Psy 320 - Cal State Northridge

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20 IQPLUS Assuming we don’t know sigma – = 0.77 – n = 25 – –We are testing at =.05 –Use Appendix D.5 Psy 320 - Cal State Northridge

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21 Appendix D.5 This table is severely abbreviated. Power for = 3.85, =.05 Psy 320 - Cal State Northridge

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22 Conclusions If we can trust our estimates in the IQPLUS study then if this study were run repeatedly, 97% of the time the result would be significant. Psy 320 - Cal State Northridge

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23 How Many Subjects Do I Really Need (Single/Related Sample(s))? Run calculations backward –Start with anticipated effect size ( ) –Determine required for power =.80. Why.80? –Calculate n What if we wanted to rerun the IQPLUS study, and wanted power =.80? Psy 320 - Cal State Northridge

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24 Calculating n We estimated =.77 Complete Appendix D.5 shows we need = 2.80 Calculations on next slide Psy 320 - Cal State Northridge

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25 IQPLUS n Psy 320 - Cal State Northridge

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26 Power for Two Independent Groups What changes from preceding? –Effect size deals with two sample means –Take into account both values of n Effect size Psy 320 - Cal State Northridge

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27 Estimating d We could calculate d directly if we knew populations. We could estimate from previous data. Here we will calculate using Violent Video Games example Psy 320 - Cal State Northridge

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28 Example: Violent Videos Games Two independent randomly selected/assigned groups –GTA (violent: 8 subjects) VS. NBA 2K7 (non- violent: 10 subjects) –We want to compare mean number of aggressive behaviors following game play –GTA had a mean of 10.25 and s = 1.669 –NBA 2K7 had a mean of 8.4 and s = 1.647 –s 2 pooled = 2.745, so s pooled = 1.657 Psy 320 - Cal State Northridge

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29 Two Independent Groups Then calculate from effect size Note: The above formula assumes that the 2 groups have equal n Psy 320 - Cal State Northridge

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30 Two Independent Groups Our data do not have equal n, but let’s pretend they do for a moment (both 10) For our data Psy 320 - Cal State Northridge

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31 Appendix D.5 This table is severely abbreviated. Power for = 2.5, =.05 Estimate =.71 Psy 320 - Cal State Northridge

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32 Conclusions If we had equal n and we can trust our estimates in the violent video game study then if this study were run repeatedly, 71% of the time the result would be significant. Psy 320 - Cal State Northridge

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33 Unequal Sample Sizes With unequal samples use harmonic mean of sample sizes Where k is the number of groups (i.e. 2), n i is each group size Standard arithmetic average will work well if n ’s are close. Psy 320 - Cal State Northridge

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34 Two Independent Groups Our data do not have equal n, so we need to find the harmonic mean For our data Psy 320 - Cal State Northridge

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35 Two Independent Groups Our data do not have equal n, so… For our data Psy 320 - Cal State Northridge

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36 Appendix D.5 This table is severely abbreviated. Psy 320 - Cal State Northridge Power for = 2.4, =.05 Estimate =.67

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37 Conclusions If we we can trust our estimates in the violent video game study then if this study were run repeatedly, 67% of the time the result would be significant. Psy 320 - Cal State Northridge

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38 How Many Subjects Do I Really Need (Independent Samples)? Run calculations backward –Start with anticipated effect size ( ) –Determine required for power =.80. Why.80? –Calculate n What if we wanted to rerun the violent video game study, and wanted power =.80? Psy 320 - Cal State Northridge

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39 Calculating n We estimated = 1.116 Complete Appendix E.5 shows we need = 2.80 Calculations on next slide Psy 320 - Cal State Northridge

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40 Violent Video Games n Psy 320 - Cal State Northridge

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